Deformations of asymptotically cylindrical associative submanifolds

TL;DR

This paper develops the deformation theory of asymptotically cylindrical associative submanifolds in G2-manifolds, establishing foundational results for their moduli spaces and potential applications in geometric gluing constructions.

Contribution

It introduces the deformation theory for ACyl associative submanifolds, analyzes their moduli spaces, and computes their virtual dimensions, enabling future geometric applications.

Findings

Moduli spaces of ACyl associative submanifolds are well-defined with natural topologies.

The virtual dimensions of these moduli spaces are computed.

In optimal cases, the moduli space embeds as a Lagrangian in the space of holomorphic curves.

Abstract

This article develops the deformation theory of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds, laying the foundation for the gluing of ACyl associative submanifolds in twisted connected sum $G_2$-manifolds presented by the author in [Ber22]. We study the moduli space of ACyl associative submanifolds with a fixed asymptotic holomorphic curve and a fixed rate, as well as the moduli space where this asymptotic data is allowed to vary, each endowed with a natural topology. We compute their virtual dimensions and show that in the best-case scenario, the latter embeds as a Lagrangian submanifold in the moduli space of holomorphic curves of the limiting Calabi-Yau 3-fold. These results may also be of independent interest for future.